# Treewidth and Minimum Fill-in of Weakly Triangulated Graphs

@inproceedings{Bouchitt1999TreewidthAM, title={Treewidth and Minimum Fill-in of Weakly Triangulated Graphs}, author={Vincent Bouchitt{\'e} and Ioan Todinca}, booktitle={STACS}, year={1999} }

We use the notion of potential maximal clique to characterize the maximal cliques appearing in minimal triangulations of a graph. We show that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum fill-in are polynomially tractable for these graphs. Finally we show how to compute in polynomial time the potential maximal cliques of weakly triangulated graphs.

## 18 Citations

Treewidth and Minimum Fill-in: Grouping the Minimal Separators

- Mathematics, Computer ScienceSIAM J. Comput.
- 2001

It is shown that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum fill-in are polynomially tractable for these graphs.

Listing all potential maximal cliques of a graph

- Mathematics
- 2019

A potential maximal clique of a graph is a vertex set that induces a maximal clique in some minimal triangulation of that graph. It is known that if these objects can be listed in polynomial time for…

Computing the Treewidth and the Minimum Fill-in with the Modular Decomposition

- Mathematics, Computer ScienceSWAT
- 2002

Using the notion of modular decomposition, it is shown that if C is a class of graphs which is modularly decomposable into graphs that have a polynomial number of minimal separators, or graphs formed by adding a matching between two cliques, then both the treewidth and the minimum fill-in problems on C can be solved inPolynomial time.

Acyclic Colorings and Triangulations of Weakly Chordal Graphs

- Mathematics
- 2009

An acyclic coloring of a graph is a proper vertex coloring without bichromatic cycles. We show that the acyclic colorings of any weakly chordal graph G correspond to the proper colorings of…

Computing the Treewidth and the Minimum Fill-In with the Modular Decomposition

- Mathematics, Computer ScienceAlgorithmica
- 2003

This work shows that if C is a class of graphs that are modularly decomposable into graphs that have a polynomial number of minimal separators, or graphs formed by adding a matching between two cliques, then both the treewidth and the minimum fill-in on C can be solved inPolynomial time.

Triangulated and Weakly Triangulated Graphs: Simpliciality in Vertices and Edges

- Mathematics
- 2001

We introduce the notion of weak simpliciality, in order to extend to weakly triangulated graphs properties of triangulated graphs, us ing Hayward’s notion that a vertexin a triangulated graph behaves…

Recognizing Weakly Triangulated Graphs by Edge Separability

- MathematicsNord. J. Comput.
- 2000

A new O(m2) recognition algorithm which is not based on the notion of a 2-pair, but rather on the structural properties of the minimal separators of the graph gives the strongest relationship to the class of triangulated graphs that has been established so far.

A revisit of the scheme for computing treewidth and minimum fill-in

- Computer ScienceTheor. Comput. Sci.
- 2014

Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure

- MathematicsArXiv
- 2022

We continue the study of (tw , ω )-bounded graph classes, that is, hereditary graph classes in which the treewidth can only be large due to the presence of a large clique, with the goal of…

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